Supersymmetry on curved spaces and superconformal anomalies

TL;DR

The paper analyzes four-dimensional SCFTs on curved manifolds preserving rigid supersymmetry and shows that the background gauge field coupled to the R-current is fixed by the Weyl tensor via the charged conformal Killing spinor equation. This leads to a dramatic simplification of the superconformal anomalies: on SUSY backgrounds the central charge c does not contribute to the trace or R-current anomalies in Lorentzian signature, and in Euclidean spaces with two opposite-R-charge supercharges the anomalies become purely topological or vanish upon integration. The authors derive the integrability conditions linking the Weyl and Maxwell sectors, present explicit backgrounds with vanishing anomalies, and connect the results to holography, where the five-dimensional minimal gauged supergravity dual reproduces the same anomaly structure through holographic renormalization and Chern–Simons terms. These findings provide a coherent framework for understanding SUSY on curved spaces and have potential implications for localization, gravity dual constructions, and higher-dimensional generalizations.

Abstract

We study the consequences of unbroken rigid supersymmetry of four-dimensional field theories placed on curved manifolds. We show that in Lorentzian signature the background vector field coupling to the R-current is determined by the Weyl tensor of the background metric. In Euclidean signature, the same holds if two supercharges of opposite R-charge are preserved, otherwise the (anti-)self-dual part of the vector field-strength is fixed by the Weyl tensor. As a result of this relation, the trace and R-current anomalies of superconformal field theories simplify, with the trace anomaly becoming purely topological. In particular, in Lorentzian signature, or in the presence of two Euclidean supercharges of opposite R-charge, supersymmetry of the background implies that the term proportional to the central charge c vanishes, both in the trace and R-current anomalies. This is equivalent to the vanishing of a superspace Weyl invariant. We comment on the implications of our results for holography.